60.2.15 problem 591
Internal
problem
ID
[10602]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
591
Date
solved
:
Tuesday, January 28, 2025 at 04:55:40 PM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
\begin{align*} y^{\prime }&=\frac {F \left (\frac {a y^{2}+b \,x^{2}}{a}\right ) x}{\sqrt {a}\, y} \end{align*}
✓ Solution by Maple
Time used: 0.288 (sec). Leaf size: 126
dsolve(diff(y(x),x) = F((a*y(x)^2+b*x^2)/a)*x/a^(1/2)/y(x),y(x), singsol=all)
\begin{align*}
y &= \operatorname {RootOf}\left (F \left (\frac {a \,\textit {\_Z}^{2}+b \,x^{2}}{a}\right ) \sqrt {a}+b \right ) \\
y &= \frac {\sqrt {a \left (-b \,x^{2}+\operatorname {RootOf}\left (a \left (\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right ) \sqrt {a}+b}d \textit {\_a} \right ) b -b \,x^{2}+2 c_{1} a \right ) a \right )}}{a} \\
y &= -\frac {\sqrt {a \left (-b \,x^{2}+\operatorname {RootOf}\left (a \left (\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right ) \sqrt {a}+b}d \textit {\_a} \right ) b -b \,x^{2}+2 c_{1} a \right ) a \right )}}{a} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.434 (sec). Leaf size: 253
DSolve[D[y[x],x] == (x*F[(b*x^2 + a*y[x]^2)/a])/(Sqrt[a]*y[x]),y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (-\frac {b K[2]}{b+\sqrt {a} F\left (\frac {b x^2+a K[2]^2}{a}\right )}-\int _1^x\left (\frac {2 b K[1] K[2] F''\left (\frac {b K[1]^2+a K[2]^2}{a}\right )}{\sqrt {a} \left (b+\sqrt {a} F\left (\frac {b K[1]^2+a K[2]^2}{a}\right )\right )}-\frac {2 b F\left (\frac {b K[1]^2+a K[2]^2}{a}\right ) K[1] K[2] F''\left (\frac {b K[1]^2+a K[2]^2}{a}\right )}{\left (b+\sqrt {a} F\left (\frac {b K[1]^2+a K[2]^2}{a}\right )\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {b F\left (\frac {b K[1]^2+a y(x)^2}{a}\right ) K[1]}{\sqrt {a} \left (b+\sqrt {a} F\left (\frac {b K[1]^2+a y(x)^2}{a}\right )\right )}dK[1]=c_1,y(x)\right ]
\]